Talk:Clifford algebra
Formulas First define: : e_1,e_2,e_3 are orthogonal unit vectors forming the standard basis and corresponding to axes x,y,z : \mathbf{\vec{a}},\mathbf{\vec{b}},\mathbf{\vec{c}},\mathbf{\vec{d}} are unit vectors: :: \mathbf{\vec{a}}=a_1e_1+a_2e_2+a_3e_3 :: \mathbf{\vec{b}}=b_1e_1+b_2e_2+b_3e_3 :: \mathbf{\vec{c}}=c_1e_1+c_2e_2+c_3e_3 :: \mathbf{\vec{d}}=d_1e_1+d_2e_2+d_3e_3 : u,v,w,x are arbitrary vectors. : M,N are arbitrary multivectors. Formulas involving the standard basis in 3 dimensions Geometric product: : \begin{align} &e_1e_2=e_1\and e_2\\ &e_1(e_1e_2)=e_2\\ &e_1(e_2e_3)=e_1e_2e_3=e_{123}=I\\ &(e_1e_2)(e_2e_3)=e_1e_2e_2e_3=e_1e_3\\ &i=-e_2e_3\\ &j=-e_3e_1\\ &k=-e_1e_2\\ &i^2=j^2=k^2=ijk=-1\\ &I^2=e_1e_2e_3e_1e_2e_3=-1\\ &e_1I=e_1e_1e_2e_3=-e_1e_2e_1e_3=e_1e_2e_3e_1=Ie_1\\ &Ie_1=e_2e_3\\ &I^2e_1=Ie_2e_3=-e_1 \end{align} Formulas involving unit vectors in 3 dimensions Wedge product: : 3 \wedge 5 = 15 : \mathbf{\vec{a}} \wedge \mathbf{\vec{a}} = 0 : \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} = bivector :: \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} = (a_2 b_3 - a_3 b_2) e_23 + (a_3 b_1 - a_1 b_3)e_31 + (a_1 b_2 - a_2 b_1)e_12 : \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} \wedge \mathbf{\vec{c}} = trivector :: \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} \wedge \mathbf{\vec{c}} = (a_1 b_2 c_3 + a_2 b_3 c_1 + a_3 b_1 c_2 - a_1 b_3 c_2 - a_2 b_1 c_3 - a_3 b_2c_1)(e_1 \wedge e_2 \wedge e_3) : \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} \wedge \mathbf{\vec{c}} \wedge \mathbf{\vec{d}} = 0 = a quadvector with zero volume : (\mathbf{\vec{a}} \wedge \mathbf{\vec{b}} \wedge \mathbf{\vec{c}}) \cdot \mathbf{\vec{c}} = \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} Since \mathbf{\vec{a}} \wedge \mathbf{\vec{a}} = 0 : 0 = (\mathbf{\vec{a}}+\mathbf{\vec{b}}) \wedge (\mathbf{\vec{a}}+\mathbf{\vec{b}}) : 0 = \mathbf{\vec{a}} \wedge \mathbf{\vec{a}} + \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} + \mathbf{\vec{b}} \wedge \mathbf{\vec{a}} + \mathbf{\vec{b}} \wedge \mathbf{\vec{b}} : 0 = 0 + \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} + \mathbf{\vec{b}} \wedge \mathbf{\vec{a}} + 0 : 0 = \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} + \mathbf{\vec{b}} \wedge \mathbf{\vec{a}} Therefore: : -(\mathbf{\vec{b}} \wedge \mathbf{\vec{a}}) = \mathbf{\vec{a}} \wedge \mathbf{\vec{b}} This means that rotation from \mathbf{\vec{b}} to \mathbf{\vec{a}} is the negative of rotation from \mathbf{\vec{a}} to \mathbf{\vec{b}} . Formulas involving bivectors in 3 dimensions For bivectors (\mathbf{\vec{a}} \wedge \mathbf{\vec{b}}) and (\mathbf{\vec{c}} \wedge \mathbf{\vec{d}}) : : (\mathbf{\vec{a}} \wedge \mathbf{\vec{b}})(\mathbf{\vec{c}} \wedge \mathbf{\vec{d}}) = (\mathbf{\vec{a}} \wedge \mathbf{\vec{b}}) \cdot (\mathbf{\vec{c}} \wedge \mathbf{\vec{d}}) + (\mathbf{\vec{a}} \wedge \mathbf{\vec{b}}) \times (\mathbf{\vec{c}} \wedge \mathbf{\vec{d}}) + (\mathbf{\vec{a}} \wedge \mathbf{\vec{b}}) \wedge (\mathbf{\vec{c}} \wedge \mathbf{\vec{d}}) Formulas involving antivectors in 3 dimensions \bar{e}_1, \bar{e}_2, \bar{e}_3 are pseudo-vectors or anti-vectors: : \bar{e}_1 = e_2 \wedge e_3 : \bar{e}_2 = e_3 \wedge e_1 : \bar{e}_3 = e_1 \wedge e_2 \bar{\mathbf{\vec{b}}} = b_1\bar{e_1} + b_2\bar{e_2} + b_3\bar{e_3} Wedge product of vector and antivector: : (a_1e_1 + a_2e_2 + a_3e_3) \wedge (b_1\bar{e}_1 + b_2\bar{e}_2 + b_3\bar{e}_3) = (a_1b_1 + a_2b_2 + a_3b_3)(e_1 \wedge e_2 \wedge e_3) = (\mathbf{\vec{a}} \cdot \mathbf{\vec{b}})I \mathbf{\vec{a}} \cdot \mathbf{\vec{b}} = a_1b_1 + a_2b_2 + a_3b_3 \mathbf{\vec{a}}\mathbf{\vec{a}} = \mathbf{\vec{a}} \cdot \mathbf{\vec{a}} The Antiwedge product \vee operates on antivectors: : \bar{e}_1 \vee \bar{e}_2 = (e_2 \wedge e_3) \vee (e_3 \wedge e_1) = e_3 : \bar{e}_2 \vee \bar{e}_3 = (e_3 \wedge e_1) \vee (e_1 \wedge e_2) = e_1 : \bar{e}_3 \vee \bar{e}_1 = (e_1 \wedge e_2) \vee (e_2 \wedge e_3) = e_2 \mathbf{\vec{c}} \vee \mathbf{\vec{d}} := ((\mathbf{\vec{c}}I^{-1}) \wedge (\mathbf{\vec{d}}I^{-1}))I Formulas involving multivectors M \wedge N := \sum_{r,s}\langle \langle M \rangle_r \langle N \rangle_s \rangle_{r+s} Use in physics When the electromagnetic field is defined as the multivector sum of an electric field vector and a magnetic field bivector, the four Maxwell equations can be reduced to a single equation.